In order to derive the explicit expressions for the two types of corrections \( \mathbf {m}^{eq,s}\;\)  and \(\mathbf {m}^{eq,G}\;\)  appearing in Eq. (REF ), we shall now perform the Chapman-Enskog multi-scale expansion [1]}, which would then allow us to recover the NS equations from the corresponding rectangular LB scheme (Eq. (REF )). In this regard, the approach used in Refs. [2]} and [3]} will be adopted. First, we expand the moment \(\mathbf {m}\)  about its equilibria \(\mathbf {m}^{(0)}\)  by including the non-equilibrium effects as a perturbation and also the time derivative \(\partial _t\)  via a multiscale time expansion as
\( \mathbf {m} =\mathbf {m}^{(0)}+\epsilon \; \mathbf {m}^{(1)} + \epsilon ^2 \; \mathbf {m}^{(2)}+ \ldots ,\qquad \partial _t= \partial _{t_0}+\epsilon \partial _{t_1} + \epsilon ^2 \partial _{t_2}+\ldots ,\) 
